The report states that Besiktas President Fikret Orman is preparing to fly to England to hold talks with Arsenal and the England international.
Welbeck could slip down to third choice striker if Arsenal sign a star striker. The Gunners have been linked with the likes of Karim Benzema over the past week by the Telegraph.
Beşiktaş recently signed Hoffenheim full-back Beck and Porto winger Ricardo Quaresma but the clubs priority remains finding a replacement for Demba Ba.
The Black Eagles sold last seasons top scorer to Shanghai Shenhua for €14 million and have yet to replace the Senegalese international.
Beşiktaş will only have two first-team strikers should they fail to sign another forward before the end of the transfer window.
Welbeck provided six assists and scored eight goals in 37 appearances for Arsenal last season.